The Limit of Humanly Knowable Mathematical Truth

The Limit of Humanly Knowable Mathematical Truth Gödel s Incompleteness Theorems, and Artificial Intelligence Santa Rosa Junior College December 12, 2015

Another title for this talk could be... An Argument Against an Argument Against Artificial Intelligence Using Gödel s Incompleteness Theorems

Another title for this talk could be... An Argument Against an Argument Against Artificial Intelligence Using Gödel s Incompleteness Theorems Using Gödel s Incompleteness Theorems

Introduction Axiomitization of Mathematics Why?

Introduction Axiomitization of Mathematics Why? Guido Grandi s Proof of God 1 1 + 1 1 + 1 1 + 1 1

Introduction Axiomitization of Mathematics Why? Guido Grandi s Proof of God 1 1 + 1 1 + 1 1 + 1 1 (1 1) + (1 1) + (1 1) + = 0

Introduction Axiomitization of Mathematics Why? Guido Grandi s Proof of God 1 1 + 1 1 + 1 1 + 1 1 (1 1) + (1 1) + (1 1) + = 0 1 + ( 1 + 1) + ( 1 + 1) + ( 1 + 1) + = 1 Thus, 1 = 0 implying we get something from nothing. Thus,...

Why is Axiomitization Necessary? Another Proof that 1 = 0 α = 1 1 2 + 1 3 1 4 + 1 5 1 6 + 1 7 1 8 +

Why is Axiomitization Necessary? Another Proof that 1 = 0 α = 1 1 2 + 1 3 1 4 + 1 5 1 6 + 1 7 1 8 + α 0.693

Why is Axiomitization Necessary? Another Proof that 1 = 0 α = 1 1 2 + 1 3 1 4 + 1 5 1 6 + 1 7 1 8 + α 0.693 1 2 α = 1 2 1 4 + 1 6 1 8 + 1 10 1 12 +

Why is Axiomitization Necessary? Another Proof that 1 = 0 α = 1 1 2 + 1 3 1 4 + 1 5 1 6 + 1 7 1 8 + α 0.693 1 2 α = 1 2 1 4 + 1 6 1 8 + 1 10 1 12 + α + 1 2 α = 1 + 1 3 2 4 + 1 5 2 8 + 1 7 +

Why is Axiomitization Necessary? Another Proof that 1 = 0 α = 1 1 2 + 1 3 1 4 + 1 5 1 6 + 1 7 1 8 + Thus, 1 2 α = 0 1 = 0 α 0.693 1 2 α = 1 2 1 4 + 1 6 1 8 + 1 10 1 12 + α + 1 2 α = 1 + 1 3 2 4 + 1 5 2 8 + 1 7 + = α

Why is Axiomitization Necessary? Calculus! Calculus was a revolutionary discovery (or invention?) by Leibniz and Newton, but at the time there were significant leaps of logic in their work. Leibiz used the notion of an infinitesimal, dx, which is a number smaller than any positive real real number to build his calculus. Mathematicians and philosophers at the time were skeptical that this new algebra of infinitesimals was logically sound. Gottfried Leibniz

Leibniz Proof of the Product Rule Leibniz said let x be a real number smaller than any other real number (a infinitesimal ). Note, in the picture below: f (x 0 + x) g(x 0 + x) = f (x 0 ) g(x 0 )+f (x 0 ) g +g(x 0 ) f + f g d dx (f g) = f (x 0 + x) g(x 0 + x) f (x 0 ) g(x 0 ) = f (x 0 ) g + g(x 0 ) f + f g = f dg dx + g df dx

Why is Axiomitization Necessary? Also,

Why is Axiomitization Necessary? Also, computers!!

Number systems axiomitized Kinda Sorta History of Number Systems The history of the discovery (or is it invention?) of number systems. N Z Q R C

Number systems axiomitized Kinda Sorta History of Number Systems The history of the discovery (or is it invention?) of number systems. N Z Q R C History of the axiomitization of number systems. C R Q Z N }{{}?

Axiomitizing the natural numbers Georg Cantor introduced the notion of sets in the late 19th century. Georg Cantor

Axiomitizing the natural numbers Georg Cantor introduced the notion of sets in the late 19th century. Georg Cantor Unfortunately, some logical inconsistencies with Cantor s (Naive) Set Theory were discovered.

Issues with Self-referencing Russell s Paradox Let X denote the set that contains all sets, except the those sets that do not contain themselves. Does X contain itself? X = {Y : Y / Y }

Issues with Self-referencing Russell s Paradox Let X denote the set that contains all sets, except the those sets that do not contain themselves. Does X contain itself? X = {Y : Y / Y } Barber of Seville The town of Seville has just one barber. This barber is a man who shaves all those, and only those, men in town who do not shave themselves. Does the barber shave himself?

Another attempt to axiomitize the natural numbers Principia Mathematica Alfred North Whitehead and Bertrand Russell published the Principia Mathematica or PM in the early 20th century that gives a set of axioms, symbols, and rules of inference from which all arithmetical truths about whole numbers could be proven...

Snippets of PM Symbols in PM is the symbol for the existential quantifier (there exists). is the symbol universal quantifier (for all). is the symbol for and, so P Q represents P and Q. is the symbol for not or negation. P Q is the symbol for the conditional statement if P then Q. There are variables for numbers w, x, y, z and variables for sentences P, Q, R, S. Punctuation is defined such as parenthesis, commas, periods, etc.

Snippets of PM Rules of Inference Modus Ponens: If P Q is a true statement in PM and P is true in PM, then Q is a true statement in PM. Modus Tollens: If P Q is a true statement in PM and Q is false in PM, then P is false in PM. Theorems and Proofs A theorem is a sentence in PM that has been proven using the axioms and rules of inferences. A proof is a finite string of sentences using the rules of inferences, axioms, and theorems.

Examples Consider the following two sentences in PM 1 2 S 1 : ( y) ( x) (y < x) S 2 : ( y) ( x) ( y = x 2)

Examples Consider the following two sentences in PM 1 S 1 : ( y) ( x) (y < x) 2 S 2 : ( y) ( x) ( y = x 2) Translation of these statements: 1 S 1 says for every whole number there is a whole number bigger than that number. (True) 2 S 2 says every real number is a perfect square. (False, y = 3)

Is PM Complete and Consistent? Is PM consistent? Suppose there was some sentence T in PM such that T and T (the negation of T ) can be proven within PM? If there exists is such a sentence, we would say that PM is inconsistent. Is PM complete? We say that PM is complete if either S or S can be proven within PM for any any sentence S that asserts some property about whole numbers.

Proving statements within the system Jumping out of the System Let T be the sentence [( n 3) ( x)( y)( z)(x n + y n = z n )].

Proving statements within the system Jumping out of the System Let T be the sentence [( n 3) ( x)( y)( z)(x n + y n = z n )]. T is Fermat s Last Theorem. It was first posed by Fermat in 1637. It was finally proven by Andrew Wiles in 1994. Wiles proof of Fermat s Last Theorem does not stay within PM.

Kurt Gödel

Richard s Paradox by Jules Richard Given a language (English) that can express purely arithmetical properties of whole numbers such as an integer is divisible by 10 and an integer is the product of two integers, etc. These properties can be placed in serial order: property a precedes property b if it has fewer letters than b or if they have the same number of letters and a precedes b alphabetically. List these properties in order, so there is a unique integer the corresponds to each property.

Richard s Paradox We say a whole number n is said to be Richardian if n does not have the property designated by its corresponding arithmetical property. Example: Suppose the property that corresponds to 17 is not divisible by any other integer other than 1 or itself. Then 17 is not Richardian. The property of whether a number is Richardian is a property of whole numbers, so it is on the list somewhere and has a corresponding number r. Is r Richardian?

Gödel s Incompleteness Theorems Gödel modeled his proof in his 1931 paper On Formally Undecidable Propositions of Principia Mathematica and Related Systems on Richard s Paradox, but he was able to circumvent the logical flaws Richard s paradox. What exactly did he do? He started with what is now called Gödel numbering.

Gödel s Incompleteness Theorems Gödel showed how to construct a formula G in PM that represents the meta-mathematical statement The formula G is not demonstrable within PM. G is called the Gödel Formula. Gödel then showed that G is demonstrable in PM if and only if G is demonstrable in PM. Thus, he showed that if PM is consistent, then it is incomplete. Gödel used a meta-mathematical argument to show that G is a true mathematical formula. Thus, PM is incomplete.

Gödel s Incompleteness Theorems Gödel also showed that PM is essentially incomplete. Suppose we take G to be a new axiom in PM. In Gödel s proof, he showed how to construct a new Gödel formula G in this new system. Moreover, his construction is recursive, so for any finitely many axioms are added to the list, a new Gödel formula can always be constructed in the new system.

Implications for Physics

Implications for Artificial Intelligence Lucas-Penrose Argument Against Artificial Intelligence Using Gödel s Incompleteness Theorems Any computer or Turing machine is built upon a finite set of instructions. If such a machine were programmable to perform operations in PM, then Gödel showed that the machine will have certain limitations that the human mind will not, such as the Gödel formula G. Such a machine can only know what it can prove using its processors, rules of inference, etc, so it will have a Gödel formula, a statement that is true, but it cannot see to be true.

Implications for Artificial Intelligence What follows is an argument against Lucas-Penrose s argument against artificial intelligence using Gödel s Incompleteness Theorems by using Gödel s Incompleteness Theorems. A single person (let s say Leonard Euler) will have a finite number of symbolic thoughts in their lifetime. Thus, the set of all symbolic mathematical truth that Euler will ever know is finite, and thus it is axiomitizable.

Limits to Humanly Knowable Mathematics Surely, Euler could do arithmetic and even prove basic statements about whole numbers. In other words the system that is Euler is strong enough to talk about arithmetic. Thus, by Gödel s Incompleteness Theorems, there is a statement about whole numbers G E that is true, but is not demonstrable by Euler. Euler has a Gödel formula. In essence, the brain or consciousness of Euler has limitations just as a Turing machine does.

Limits to Humanly Knowable Mathematics The number of symbolic thoughts of any one person is finite. Thus, at any moment in history, the total number of symbolic thoughts by all humans is still finite and hence axiomitizable. Thus, humanity as a whole (at any moment in time) has a Gödel formula, G H, a statement about whole numbers that while true, we can never reach.

Thank you! Questions? Comments? Discussion?

Bibliography By request. I do have one.